A box 'A' contains 2 white, 3 red and 2 black balls.

Question

A box 'A' contains $2$ white, $3$ red and $2$ black balls. Another box 'B' contains $4$ white, $2$ red and $3$ black balls. If two balls are drawn at random, without eplacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red,  then the probability that both balls are drawn from box 'B' is :

Accepted Answer

Probability of drawing a white ball and then a red ball

from bag  B is given by
$${{{}^4{C_1} \times {}^2{C_1}} \over {{}^9{C_2}}}$$ = $${2 \over 9}$$

Probability of drawing a white ball and then a red ball

from bag A is given by $${{{}^2{C_1} \times {}^3{C_1}} \over {{}^7{C_2}}}$$ = $${2 \over 7}$$

Hence, the probability of drawing a white ball and then

a red ball from bag B = $${{{2 \over 9}} \over {{2 \over 7} + {2 \over 9}}}$$ = $${{2 \times 7} \over {18 + 14}}$$ = $${7 \over {16}}$$

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